I mapped out the structure of the board as a graph of which squares you can get to from which other squares. It's surprisingly stringy without all the squares forbidden by the queen placement.
It made a lot more sense after doing that, as I'm nowhere near proficient enough at chess to see the board and all its moves in my mind's eye.
Based on the graph, I also found an optimal route through the whole board: https://gist.github.com/unflxw/b4d21c55c137a9a42fba75bd0cf98...
I'm used to a version with pawns that involves less backtracking, so it took me a little while to stop trying to be "smarter" and just be okay with continually returning to home base.